Geodesic
How to define "convex functions" on differentiable manifolds
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 29 (2009) no. 1, pp. 7-17

Voir la notice de l'article provenant de la source Library of Science

In the paper a class of families (M) of functions defined on differentiable manifolds M with the following properties:
Keywords: Fréchet differetiability, Gateaux differentiability, locally strongly paraconvex functions, $C^{1,u}$-manifolds 
Rolewicz, Stefan. How to define "convex functions" on differentiable manifolds. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 29 (2009) no. 1, pp. 7-17. http://geodesic.mathdoc.fr/item/DMDICO_2009_29_1_a0/
@article{DMDICO_2009_29_1_a0,
     author = {Rolewicz, Stefan},
     title = {How to define "convex functions" on differentiable manifolds},
     journal = {Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
     pages = {7--17},
     year = {2009},
     volume = {29},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMDICO_2009_29_1_a0/}
}
TY  - JOUR
AU  - Rolewicz, Stefan
TI  - How to define "convex functions" on differentiable manifolds
JO  - Discussiones Mathematicae. Differential Inclusions, Control and Optimization
PY  - 2009
SP  - 7
EP  - 17
VL  - 29
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DMDICO_2009_29_1_a0/
LA  - en
ID  - DMDICO_2009_29_1_a0
ER  - 
%0 Journal Article
%A Rolewicz, Stefan
%T How to define "convex functions" on differentiable manifolds
%J Discussiones Mathematicae. Differential Inclusions, Control and Optimization
%D 2009
%P 7-17
%V 29
%N 1
%U http://geodesic.mathdoc.fr/item/DMDICO_2009_29_1_a0/
%G en
%F DMDICO_2009_29_1_a0

[1] [1] E. Asplund, Farthest points in reflexive locally uniformly rotund Banach spaces, Israel Jour. Math. 4 (1966), 213-216.

[2] [2] E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31-47.

[3] [3] S. Lang, Introduction to differentiable manifolds, Interscience Publishers (division of John Wiley Sons) New York, London, 1962.

[4] [4] S. Mazur, Über konvexe Mengen in linearen normierten Räumen, Stud. Math. 4 (1933), 70-84.

[5] [5] E. Michael, Local properties of topological spaces, Duke Math. Jour. 21 (1954), 163-174.

[6] [6] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Springer-Verlag, 1364 (1989).

[7] [7] D. Preiss and L. Zajíček, Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions, Proc. 11-th Winter School, Suppl. Rend. Circ. Mat di Palermo, ser II, 3 (1984), 219-223.

[8] [8] S. Rolewicz, On α(·)-monotone multifunction and differentiability of γ-paraconvex functions, Stud. Math. 133 (1999), 29-37.

[9] 9[] S. Rolewicz, On α(·)-paraconvex and strongly α(·)-paraconvex functions, Control and Cybernetics 29 (2000), 367-377.

[10] [10] S. Rolewicz, On the coincidence of some subdifferentials in the class of α(·)-paraconvex functions, Optimization 50 (2001), 353-360.

[11] [11] S. Rolewicz, On uniformly approximate convex and strongly α(·)-paraconvex functions, Control and Cybernetics 30 (2001), 323-330.

[12] [12] S. Rolewicz, α(·)-monotone multifunctions and differentiability of strongly α(·)-paraconvex functions, Control and Cybernetics 31 (2002), 601-619.

[13] [13] S. Rolewicz, On differentiability of strongly α(·)-paraconvex functions in non-separable Asplund spaces, Studia Math. 167 (2005), 235-244.

[14] [14] S. Rolewicz, Paraconvex analysis, Control and Cybernetics 34 (2005), 951-965.

[15] [15] S. Rolewicz, An extension of Mazur Theorem about Gateaux differentiability, Studia Math. 172 (2006), 243-248.

[16] [16] S. Rolewicz, Paraconvex Analysis on $C^{1,u}_E$-manifolds, Optimization 56 (2007), 49-60.

[17] [17] L. Zajicěk, Differentiability of approximately convex, semiconcave and strongly paraconvex functions, Jour. Convex Analysis 15 (2008), 1-15.